Imaging inverse problems such as reconstruction, deblurring, super-resolution, inpainting, or denoising can be solved by assuming the target image is sparse in the wavelet domain. Orthogonal wavelet transforms are not shift-invariant; image features are processed differently depending on their position with respect to the wavelet grid, and this can result in blocking artifacts.
Techniques to solve the proximal operation performed during reconstruction have been developed, such as the utilization of an undecimated redundant wavelet transforms in conjunction with a nested inner algorithm inside the main solver (e.g. Chambolle-Pock or Dykstra inside FISTA).
Dynamic contrast-inhanced (DCE) MRI involves injection of a contrast agent (CA) into the patient and observing the dynamic intake of the CA into tissue at several different points in time using MRI. Image reconstruction methods apply multiple shifted wavelet transforms at each iteration, which incurs an extra computational cost of at least a factor of 2 per image dimension. Additionally, exact undecimated wavelet methods also store the wavelet coefficients of the image, which are larger than the image by a factor of two in every dimension. This becomes very large with large-dimension use-cases such as dynamic volumetric reconstruction (3D+T), motion-compensated imaging, multi-contrast imaging including flow or diffusion, etc.